I NTRODUCTION TO R OBOTICS CPSC Lecture 4B – Computing the Jacobian

R EMEMBER DH PARMETER The transformation matrix T

The i th column of J v is given by: The i th column of J  is given by: J ACOBIAN M ATRIX

2-DOF planar robot arm Find: Jacobian Here, n=2, 22 11 (x, y) l2l2 l1l1

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J ACOBIAN M ATRIX 2-DOF planar robot arm 22 11 (x, y) a2a2 a1a1

J ACOBIAN M ATRIX

The required Jacobian matrix J

The DH parameters are: S TANFORD MANIPULATOR

T4 = [ c1c2c4-s1s4, -c1s2, -c1c2s4-s1*c4, c1s2d3-sin1d2] [ s1c2c4+c1s4, -s1s2, -s1c2s4+c1c4, s1s2d3+c1*d2] [-s2c4, -c2, s2s4, c2*d3] [ 0, 0, 0, 1] S TANFORD MANIPULATOR

T5 = [ (c1c2c4-s1s4)c5-c1s2s5, c1c2s4+s1c4, (c1c2c4-s1s4)s5+c1s2c5, c1s2d3-s1d2] [ (s1c2c4+c1s4)c5-s1s2s5, s1c2s4-c1c4, (s1c2c4+c1s4)s5+s1s2c5, s1s2d3+c1d2] [ -s2c4c5-c2s5, -s2s4, -s2c4s5+c2c5, c2d3] [ 0, 0, 0, 1] S TANFORD MANIPULATOR

T5 = [ (c1c2c4-s1s4)c5-c1s2s5, c1c2s4+s1c4, (c1c2c4-s1s4)s5+c1s2c5, c1s2d3-s1d2] [ (s1c2c4+c1s4)c5-s1s2s5, s1c2s4-c1c4, (s1c2c4+c1s4)s5+s1s2c5, s1s2d3+c1d2] [ -s2c4c5-c2s5, -s2s4, -s2c4s5+c2c5, c2d3] [ 0, 0, 0, 1] S TANFORD MANIPULATOR

T6 = [ c6c5c1c2c4-c6c5s1s4-c6c1s2s5+s6c1c2s4+s6s1c4, - c5c1c2c4+s6c5s1s4+s6c1s2s5+c6c1c2s4+c6s1c4, s5c1c2c4-s5s1s4+c1s2c5, d6s5c1c2c4-d6s5s1s4+d6c1s2c5+c1s2d3-s1d2] [ c6c5s1c2c4+c6c5c1s4-c6s1s2s5+s6s1c2s4-s6c1c4, -s6c5s1c2c4- s6c5c1s4+s6s1s2s5+c6s1c2s4-c6c1c4, s5s1c2c4+s5c1s4+s1s2c5, d6s5s1c2c4+d6s5c1s4+d6s1s2c5+s1s2d3+c1d2] [ -c6s2c4c5-c6c2s5-s2s4s6, s6s2c4c5+s6c2s5-s2s4c6, -s2c4s5+c2c5, - d6s2c4s5+d6c2c5+c2d3] [ 0, 0, 0, 1] S TANFORD MANIPULATOR

Joints 1,2 are revolute Joint 3 is prismatic The required Jacobian matrix J S TANFORD MANIPULATOR

The relation between the joint and end-effector velocities: where J(m×n). If J is a square matrix (m=n), the joint velocities: If m<n, we use pseudoinverse J + where I NVERSE V ELOCITY

A CCELERATION The relation between the joint and end-effector velocities: Differentiating this equation yields an expression for the acceleration: Given of the end-effector acceleration, the joint acceleration

S INGULARITIES J is a function of joint angles q => J(q) The columns of J form a basis for the space of possible end effector velocities Rank (J) <= min(6,n), when rank(J) = n “Full Rank) But the rank of the Jacobian is not necessarily constant… it will of course depend upon the configuration / pose of the manipulator Configurations at which the rank of J decreases are special configuration called Singular configurations or simply singularities Definition: we say that any configuration in which the rank of J is less than its maximum is a singular configuration i.e. any configuration that causes J to lose rank is a singular configuration

S INGULARITIES Can we characterize how close we are to a singularity? Can’t move much this way Can move a lot this way

Jacobian Singularities: Example The four singularities of the three-link planar arm:

C HARACTERISTICS OF S INGULARITIES Singularities represent configurations from which certain directions of motion may be unattainable. At Singularities, bounded end-effector velocities may correspond to unbounded joint velocities At singularities, bounded end-effector forces/torques may correspond to unbounded joint torques Singularities usually (but not always) correspond to points on the boundary of the workspace, that is, points of maximum reach of the manipulator Singularizes correspond to points the manipulator workspace that might be unreachable by small perturbations Near singularities there will not exist a unique solution to the inverse kinematics problem. There may be no solution or infinite solutions.